The modified upward-closedness conjecture for Wakamatsu tilting modules

Let $\Lambda$ be an artin algebra, and let $T,U\in\operatorname{\mathsf{W\mathchar`-tilt}}\Lambda$. Let $\mathcal{X}_U$ and $\mathcal{Y}_T$ denote the subcategories defined in the paper.

Modified upward-closedness conjecture. If $T\in\mathcal{X}_U$ and $U\in\mathcal{Y}_T$, and $U$ is tilting, then $T$ is tilting.

This is proposed as an analogue of the upward-closedness theorem for representation-finite algebras, because for representation-infinite algebras the relation $T\geq U$ alone does not imply the two stated subcategory-membership conditions.